2 Table of Distributions

Chapter Preview. This table of distributions is a summary of selected discrete and continuous probability distributions used throughout Loss Data Analytics.

\[ {\small \begin{matrix} \begin{array}{l|cccccc} \hline \text{Name} & \text{Probability Density} & \text{Mean} & \text{Variance } \sigma^2 & \text{Moments } \mu_k'=\mathrm{E~}X^k & \mathrm{E~}(X\wedge x)^k & \text{Moment Generating} \\ & \text{Function f(x)} & \mu=\mathrm{E~}X & \mathrm{E~}(X-\mu)^2 & \text{or } \mu_k=\mathrm{E~}(X-\mu)^k & & \text{Function M(t)}=\mathrm{E~}e^{tX} \\ \hline \text{Uniform} & \frac{1}{\beta-\alpha} & \frac{\beta+\alpha}{2} & \frac{(\beta-\alpha)^2}{12} & \mu_k=0 \text{ for k odd} & & \frac{e^{\beta t}-e^{\alpha t}}{(\beta-\alpha)t} \\ & -\infty<\alpha, <\beta<\infty & & & \mu_k=\frac{(\beta-\alpha)^k}{2^k(k+1)} \text{ for k even} & & \\ \hline \text{Normal} & \frac{1}{\sqrt{2\pi}\sigma}\exp\left( -\frac{(x-\mu)^2}{2\sigma^2}\right) & \mu & \sigma^2 & \mu_k=0 \text{ for k odd} & & \exp(\mu t+\sigma^2~t^2/2) \\ & -\infty<\mu<\infty, \sigma>0 & & & \mu_k=\frac{k!\sigma^k}{(k/2)!2^{k/2}} \text{ for k even} & & \\ \hline \text{Exponential} & \frac{1}{\theta}e^{-x/\theta} & \theta & \theta^2 & \mu_k'=\theta^k \Gamma (k+1) & \theta^k\Gamma (k+1)\Gamma (k+1;x/\theta) & \frac{1}{1-\theta~t} \\ & \lambda>0 & & & & +x^k e^{-x/\theta} & \\ \hline \text{Gamma} & \frac{1}{\theta^{\alpha}\Gamma(\alpha)}x^{\alpha-1}e^{-x/\theta} & \alpha~\theta & \alpha~\theta^2 & \mu_k'=\frac{\theta^k\Gamma(k+\alpha)}{\Gamma(\alpha)} & \frac{\theta^k\Gamma(k+\alpha)}{\Gamma(\alpha)}\Gamma(k+\alpha; x/\theta) & \frac{1}{(1-\theta t)^{\alpha}} \\ & \theta>0, \alpha>0 & & & & +x^k[1-\Gamma(\alpha; x/\theta)] & \\ \hline \text{Beta} & \frac{1}{B(a,b)} u^a(1-u)^{b-1}\frac{\theta}{x}, & \frac{a \theta}{a+b} & \frac{ab\theta^2}{(a+b+1)(a+b)^2} & \mu_k'=\theta^k \frac{B(k+a,b)}{B(a,b)} & \text{Not useful} & \text{Not useful} \\ & u=x/\theta , a>0 , b>0 & & & & & \\ \hline \text{Cauchy} & \frac{1}{\pi\beta}[1+\left( \frac{x-\alpha}{\beta}\right)^2]^{-1} & \text{Does not} & \text{Does not exist} & \text{Does not exist} & \text{Does not exist} & \text{Does not exist} \\ & -\infty <\alpha <\infty, \beta>0 & \text{exist} & & & & \\ \hline \text{Lognormal} & & \exp(\mu+ & \exp(2\mu +2\sigma^2)- & \mu_k'=\exp(k\mu+k\sigma^2) & \exp(k\mu+k\sigma^2) & \text{Not useful} \\ & \frac{1}{x\sqrt{2\pi}\sigma}\exp\left( -\frac{(\ln x-\mu)^2}{2\sigma^2}\right) & \sigma^2/2) & \exp(2\mu+\sigma^2) & & \Phi\left( \frac{lnx-\mu-k\sigma^2}{\sigma}\right) & \\ & -\infty <\mu <\infty, \sigma>0 & & & & +x^k(1-F(x)) & \\ \hline \text{Pareto} & \frac{\alpha \theta^{\alpha}}{x^{\alpha+1}}, \alpha>0 & \frac{\alpha\theta}{\alpha-1} & \frac{\alpha\theta^2}{\alpha-2}-\left( \frac{\alpha \theta}{\alpha-1}\right)^2 & \mu_k'=\frac{\alpha\theta^k}{\alpha-k} & \frac{\alpha\theta^k}{\alpha-k}-\frac{k\theta^{\alpha}}{(\alpha-k)x^{\alpha-k}} & \text{Does not exist} \\ \hline \end{array} \end{matrix} } \]